Electron Spin — Question 2

Question 2

Complex spinor coefficients determine z outcomes through their moduli

In the orthonormal z basis an electron has $|\chi\rangle=C[2|+z\rangle+(1+i)|-z\rangle]$, with C real and positive.

1. Normalize the spinor and find both z probabilities.

2. Calculate the mean and standard deviation of $S_z$.

Question 2 - Solution

Complex spinor coefficients determine z outcomes through their moduli

(a) Method and calculation

$$1=C^2(4+|1+i|^2)=6C^2\Rightarrow\boxed{C=1/\sqrt6}.$$$$\boxed{P_+=2/3,\quad P_-=1/3}.$$

(b) Interpretation and further calculation

$$\langle S_z\rangle=\tfrac\hbar2(P_+-P_-)=\boxed{\hbar/6},\quad\langle S_z^2\rangle=\hbar^2/4.$$$$\boxed{\Delta S_z=\sqrt{\hbar^2/4-\hbar^2/36}=\sqrt2\hbar/3}.$$

Physical check

A complex amplitude is squared using its modulus. Every individual z result remains plus or minus hbar/2; the mean need not be an allowed individual result.